An operator approach to nonlinear parabolic type problems with variable transformation

A. A. Kornuta ; V. A. Lukyanenko

Taurida Journal of Computer Science Theory and Mathematics, no. 2 (2024), pp. 26-56

Résumé

The article deals with an initial-boundary value problem for a nonlinear functional-differential equation (FDE) of parabolic type with transformation of the unknown function arguments. It describes the phenomenon of structure formation in an optical system, consisting of a thin layer of a nonlinear Kerr-type medium and a differently organized two-dimensional feedback external contour. The dynamics of the system depends on the change in the parameters of the optical signal at the input or the parameters of the system itself. The research takes into account the abstract operator model of the initial-boundary value problem with the transformation of spatial variables. The issue considers the occurrence of spatially inhomogeneous secondary modes as a result of bifurcations, accompanying the loss of stability of a spatially homogeneous equilibrium. The authors of the article proved the discreteness of the linear operator spectrum. They investigated the nature of stability depending on the bifurcation parameters. The nonzero neutral eigenvalues are conjugate in complex for the commutative case. The condition for the existence of a neutral spectrum determines a neutral line on the plane of bifurcation parameters. Transversal intersections of the neutral line within changing the parameters along a certain path, lead to a transversal intersection of the imaginary axis by a branch of eigenvalues, that is, to the emergence (disappearance) of an unstable linear mode. The latter oscillates when the real part of the eigenvalue is equal to zero, and oscillatory instability occurs. The specifications are given for the modeled problem for a ring, circle, circumference, strip, and rectangle. The results are also represented, showing the applicability of the semigroup theory and the central manifold method. The sectoral segment of the linear problem operator is carried out, and the theorem on the central manifold is proved in the article. The proof of the applicability of the central manifold to the initial-boundary value problems for nonlinear FDEs of parabolic type, considered in the article, allows us to specify the secondary modes, taking into account the domain, eigenfunctions of the corresponding spectral problem, and the type of transformation of spatial variables. The results of the conducted analysis of the abstract equation are applicable to the study of the emergence of equilibria instability and excitation of secondary modes motion of dynamic systems, generated by various equations of the reaction-diffusion type with non-local kinetics. The latter depends on the interaction between the «substrate» (medium) filling the considered and shifted elementary volumes, where the shift is determined by a given diffeomorphism of the region filled with the «substrate». The research highlights the characteristic formalized mathematical model with power of nonlinearity. It is an initial-boundary value problem with Neumann conditions and a periodicity condition for parabolic-type FDEs with transformation of spatial variables.

Détail

Keywords: functional differential equation, spatially homogeneous equilibrium, neutral spectrum, central manifold.
Mots-clés : bifurcation